Question

PROBLEM SOLVING A study in Florida found that the number \(s\) of fish species in a pool or lake can be modeled by the function $$ s=30.6-20.5 \log A+3.8(\log A)^2 $$ where \(A\) is the area (in square meters) of the pool or lake. a. Use a graphing calculator to graph the function on the domain \(200 \leq A \leq 35,000\). b. Use your graph to estimate the number of species in a lake with an area of 30,000 square meters. c. Use your graph to estimate the area of a lake that contains six species of fish. d. Describe what happens to the number of fish species as the area of a pool or lake increases. Explain why your answer makes sense.

Short answer

A graphing calculator is used to graph the function and make estimates. The exact number of species and area to expect would vary depending on the exact values on the graph. However, the general trend appears to be that as the area of a pool or lake increases, the number of fish species also increases, though the rate of increase may vary. This makes sense, as larger bodies of water would generally be able to support more fish species.

Step-by-step solution

Graph the Function

Using a graphing calculator, input the function \( s = 30.6 - 20.5\log(A) + 3.8(\log(A))^2 \). Set the domain as \( 200 \leq A \leq 35000 \) and observe the resulting graph.

Estimate the Species Number for a Given Area

Heading back to the graph, look for the point where A equals 30,000. The y-coordinate of this point would indicate the estimated number of species in a lake with an area of 30,000 square meters.

Estimate the Area for a Given Number of Species

Using the same graph, now look for the point(s) where the function s equals 6. The x-coordinate of this/these point(s) gives the estimated area of a lake that contains six species of fish.

Describe the Trend and Give an Explanation

Record the pattern of the graph as the area (A) increases. The description should summarize the trend of fish species in relation with increasing area of a pool or a lake.

Key concepts

Graphing Calculator Usage

Graphing calculators are essential tools for students and professionals in various scientific and mathematical fields. When dealing with complex functions, such as the logarithmic function in the given exercise, a graphing calculator simplifies the process of visualizing and solving problems.

For instance, to graph the function representing fish species in a body of water, we input the mathematical model, define the domain, and use the built-in graphing features to plot the function. This visual aid is crucial for interpreting relationships between variables and finding specific values. Using a graphing calculator, students can confidently tackle part a of the exercise, which requires graphing the function on the specified domain. Furthermore, it facilitates the estimation tasks in parts b and c by allowing students to read off the values directly from the graph.

By mastering the use of a graphing calculator, students can not only complete their textbook exercises effectively but also develop skills that are valuable in real-world applications where data visualization and analysis are required.

Modeling Biodiversity

Biodiversity within ecosystems is a cornerstone of environmental science, and mathematical models can be used to represent complex natural relationships. The exercise provides a practical example of this with a logarithmic function modeling the relationship between the area of lakes and the number of fish species.

Modeling with logarithmic functions, such as the one provided (\( s = 30.6 - 20.5\text{log}(A) + 3.8(\text{log}(A))^2 \) where \( s \) is the number of species and \( A \) is the area), helps to capture the rate at which biodiversity changes with habitat size. In reality, such models are derived from empirical data and can be used to make predictions or inform conservation strategies. For students completing part d, understanding the effects of area on biodiversity is important. As they describe the trend of the number of fish species, they delve into practical conservation biology, gaining insights into how space limitations can affect species richness.

The process of creating and analyzing such models is a valuable skill set for students interested in fields like ecology, environmental policy, and natural resource management.

Interpreting Function Graphs

Interpreting graphs of functions is an important analytical skill that extends beyond the classroom. In the context of the biodiversity exercise, interpreting the graph helps students understand how fish species vary with lake area. From the function's graph, students can observe the shape and pattern, discerning the rate at which species increase or decrease as area changes.

Consider the trend analysis in part d. By studying the function's curvature, students can predict that at a certain point, increases in area yield diminishing returns in species diversity. This interpretation reflects real-world ecological principles such as habitat saturation. Such insights are not only academically enriching but also nurture critical thinking skills. Interpreting function graphs teaches students to analyze quantitative relationships, identify trends, and make data-driven predictions, which are invaluable skills in many professional fields, including economics, engineering, and environmental sciences.